To the reader:
For many years Richard Hoagland has been the most conspicuous proponent
for the idea that the Face on Mars and various other formations in the
Cydonia region of that planet are artificial objects constructed by an
intelligent civilization that once lived there. Until the summer of 1996
I was completely unaware of this controversial issue. After hearing
Hoagland discuss this topic on the Art Bell Show, I decided to take a close
look at the evidence presented in his book The Monuments of Mars
. This includes photographs (which come from the Viking mission in
the 1970s) showing some formations, such as the Face and the D&M Pyramid,
which do indeed look rather unusual. However, in addition to this
photographic evidence, Hoagland discusses at considerable length various
mathematical relationships, which he seems to regard as strong evidence
for artificiality. But my intuition as a mathematician told me that
the kinds of mathematical relationships which he describes are not at all
remarkable. I then decided to do a simple numerical experiment with randomly
chosen numbers. The results confirmed that my intuition was very well-founded
and so, from my perspective, the mathematical evidence presented in Hoagland's
book is quite fallacious and hardly worthy of being presented as serious
evidence.
In the following rather long essay I attempt to give a detailed explanation
of why I find this mathematical evidence fallacious. At times, as I was
writing this essay, I wondered if it is really worth the time and trouble.
Hoagland never makes any serious attempt to justify his professed belief
that all of the mathematical relationships for the D&M Pyramid that
he presents have any merit as scientific evidence. Concerning various
"alignments" that he and others have noticed, Hoagland offers some probability
arguments which are completely "a posteriori" and therefore have
no value as scientific evidence. His thinking on this whole issue seems
incredibly shallow to me. Is it really worthwhile responding to that?
Quite frankly, I don't know if it is worthwhile. My efforts
over the past several years to challenge Hoagland on this issue seem to
have accomplished very little. In the Fall of 1996, I gave Hoagland
a summary of the numerical experiment mentioned above. That simple experiment
demonstrated how easily one can find striking patterns of mathematical
relationships like the ones that he presents in his book. This was an attempt
on my part to convince him that scientists were quite right to reject that
kind of evidence. After that, I wrote to him several times, but he
never responded. Later, I wrote to Stanley McDaniel, thinking that
he might be able to pursuade Hoagland to respond. I also persistently
made my objections known to Art Bell. Several times Bell suggested the
possibility of a debate. I told him that I would be willing, but such a
debate never did happen. I imagine that Hoagland simply refused.
So instead, I sent several faxes to Art Bell on this topic, which
explained my point of view in a simplified way, but these were never read
on his show. As I mention in the essay below, I eventually decided
to directly challenge Hoagland to a debate. I sent a copy of my challenge
to Art Bell and a number of his frequent guests. To my surprise,
Art Bell confronted Richard Hoagland on the air with my challenge, but
he again refused.
Richard Hoagland seems to be quite attached to all of the mathematical
relationships that he presents in his book. He has mentioned them many
times on the Art Bell Show. In his public lectures, he shows slides illustrating
dozens of triangles formed by taking various points in the Cydonia region
as vertices and also listing various numerical relationships involving
ratios of angles from the D&M Pyramid. All of this is prominently
displayed (in four colors) on a poster which is available for purchase
on The Enterprise Mission website. I have come to believe that this
collection of mathematical relationships, which Hoagland refers to as the
"Geometry of Cydonia," is really just a propaganda tool for him -
one which he has apparently found to be rather effective. I seriously
doubt the sincerity of his professed belief. But if I am wrong,
let him respond to all of the objections that I have raised in the following
essay. I again challenge him to justify his belief that this evidence
has any validity and should not just be dismissed as
pseudoscientific nonsense.
In my first letter to Hoagland (dated November 16th, 1996), I included a summary of my credentials as a mathematician, explaining that I was doing that so that he would take my opinions seriously. Readers who want to know more about me can just go to
my homepage. There is a link to my curriculum
vita which
provides information about my background.

THE D&M PYRAMID
ON MARS

INTRODUCTION:
There is some kind of formation in the Cydonia region of Mars which has
been called the D & M Pyramid. Richard Hoagland named it after
V. DiPietro and G. Molenaar and also proposed the hypothesis that this
formation may have been a bilaterally-symmetric, five-sided pyramid
at one time in the past. Erol Torun proposed specific values for
various angles which are formed by the corners of the base and a point
inside the base lying directly below the apex of this hypothetical pyramid.
For more background information, see Erol Torun's article The
D&M Pyramid of Mars.

Torun found a number of numerical relationships exhibited
by these angles. They can be found in his article. Richard
Hoagland lists these relationships and some more in his book The Monuments
of Mars and also on the following page from his website:
Cydonia
Geometric Relationship Model.Here
are four of those relationships:

C/A
~\/2 ,
B/D
~
\/3
, C/F
~ \/3
, A/D
~ e/pi

To explain the notation, if a and b are positive
numbers, then a ~
b will mean that a and b are approximately equal. The error
will always be expressed as a percentage, namely the difference
between a and b expressed as a percentage of the number b. For example,
one might write 199..~..200.
The difference is 1 and so the error is .5% because 1 is .5% of 200.
The percentage errors for the above four approximations are
.5%, .17%, .7%, .08% , respectively. The
percentage errors in the numerical relationships listed by Hoagland are
in the range .05% to 1.6%, with most of them in the range .1%
to .5%. In what follows, the numerical relations listed in
Hoagland's book will be referred to as the "Torun-Hoagland Model."

ADDITIONAL RELATIONSHIPS: We will give a
number of additional numerical relationships exhibited by the above eight
angles which have a similar level of accuracy to those listed by Hoagland.
These were either overlooked or ignored by Hoagland and Torun. Some
of these new relationships fit into nice sequences with some of those from
the Torun-Hoagland Model. The percentage errors will be given in
parentheses. For comparison, one can find a tabulation of the
percentage errors for all of the numerical relationships given in The
Monuments of Marshere.

Ratios of square roots.
Here is a set of six relationships involving ratios of the above angles
and ratios of consecutive terms in the sequence of square roots of positive
integers: \/1
, \/2,
\/3 ,
\/4,
\/5,
\/6 ,
\/7 .

The first of these approximations is one of the
relationships in the Torun -Hoagland Model. The percentage errors (indicated
in parentheses) are well within the range which Hoagland and Torun allow
for the relationships which they found. Also , it should be
pointed out that these relationships are not independent. For example,
multiplying the relationships involving F/G and H/F will give the one involving
H/G.

More square roots. The first two numerical
relationships of the Torun-Hoagland Model are the beginning of another
nice sequence of approximations. But another angle from the D&M
pyramid is needed which Hoagland denotes by the letter I and which
has the value I = 100.4o.

These relationships should be compared to some of the
relationships in theTorun-Hoagland Model , namely those which involve
\/3.
It just so happens that \/3
~
1/gamma with an error of about
.025%. Therefore, any relationship involving the number \/3.can
be replaced by one which involves the number 1/gamma.
The
percentage error will change slightly.

Logs. Here are a couple of relations
involving natural logarithms.

D/I ~
ln(2)
(.3%)

F/G ~
ln(3)
(.1%)

CRITIQUE: I have
given so many additional relationships just to make the point that with
the rather generous level of accuracy that is allowed by Hoagland and Torun,
it is fairly easy to find a variety of striking numerical relationships.
This same point will be made on another page involving a "Telephone
Number Experiment" which I did in the Fall of 1996. That
experiment was an attempt on my part to convince Richard Hoagland that
one should not put any significance on the existence of such relationships.
In his book, his public lectures, and his frequent appearances on the Art
Bell Show, he has often talked about the "Geometry of Cydonia," describing
it as an exquisite set of mathematical relationships exhibited throughout
the Cydonia region. He claims that these mathematical relationships contain
some kind of message which is related to the geometry of a regular tetrahedron
inscribed in a sphere and related furthermore to what he calls "hyperdimensional
physics." On the basis of this claim, he argues that this set
of relationships must be part of an intentional design and therefore constitutes
strong evidence that various formations, including the D&M Pyramid
and the Face on Mars, are artificial.
Undoubtedly humans will eventually visit Mars and have an opportunity to
take a close look at the D&M Pyramid, the Face, and some of the other
formations in that region. It then may be obvious whether or not they are
artificial. At the present time, we only have the photographs taken
by the Viking Orbiter in 1976 and by the Mars Global Surveyor since 1998.
These have been extensively examined by various people. (Here is
a small collection of links.) Concerning
the D&M Pyramid, it is just a conjecture that this object was at one
time a symmetrically-shaped, five-sided pyramid. It is not even clear
that it ever had five well-defined corners or a true apex. By themselves,
the photographs of the D&M Pyramid (which date from 1976) don't seem
to me to provide a good case for the artificiality of that object.
But do the mathematical relationships that were noticed by Erol Torun and
Richard Hoagland strengthen the case? In my opinion, the answer is no!
I made my objections known to Art Bell and he told me several times that
he would try to arrange a debate between myself and Richard Hoagland. But
it didn't happen and so I decided in May, 1998 to directly challenge Hoagland
to a debate. He never responded to my letter, but on May 26th, to my surprise,
Art Bell confronted Hoagland on the air with my challenge.
Hoagland refused my challenge and chose to be rather impolite and insulting
in his refusal. Some of my acquaintances who listened to him thought
that he seemed terrified of the prospect of debating me. Although
I believed that my mathematical arguments were quite strong, I felt rather
terrified myself because I had absolutely no experience in such a confrontational,
debating situation. Nevertheless, I tried for several months
after that to pursuade Art Bell to let me have just fifteen minutes on
the air in order to briefly explain my point of view and to put more pressure
on Hoagland to accept my challenge to a debate, but to no avail.
In any case, here are the main points that I would try to make in such
a debate.

1.
Hoagland and Torun have given themselves too much freedom in finding
the relationships that they present. These relationships are just
approximations and the level of accuracy that Hoagland and Torun accept
is rather generous. Perhaps they believed that allowing such a level
of accuracy was justified because of the fact that the angles themselves
could not be estimated very accurately. No matter what their reason,
the existence of many such relationships is not at all surprising, as I've
already pointed out above. My telephone number experiment was intended
to show that even if eight angles were chosen in a random way, one could
still find many striking relationships involving those angles and with
a level of accuracy similar to that in the Torun-Hoagland Model. The reader
should make a close comparison of the percentage errors in my telephone
number experimentwith those occurring in
the relationships found by Torun and Hoagland (which I have tabulated..here).
The percentage errors vary considerably in both cases, but the overall
level of accuracy is quite comparable.One
way to view the matter is as follows. To start with, there are the
nine angles A, B, . . . , I. One can then compute the ratios formed
by these angles. This would give 72 numbers (excluding the ratios A/A,
B/B which are equal to 1). In addition, one can evaluate the trigonometric
functions sin, cos, and tan at those nine angles. This gives 27 more
numbers. One can also include the radian values for those angles,
which gives 9 more numbers. Altogether, one has a collection of more than
one hundred numbers. They turn out to be mostly in the range between 0
and 4 and more than half of them are in the range between 0 and 1. On the
other hand, there are certainly many important mathematical constants which
occur in the same ranges, forming another collection of numbers. In effect,
one has two large collections of numbers crowded into a fairly small interval.
That some numbers from the first collection turn out to be close approximations
to some numbers from the second collection is inevitable. The number of
such occurrences depends on how close one requires the approximations to
be.
In The Monuments of Mars, Richard Hoagland dismisses the possibility
that the numerical relationships and alignments that he refers to as the
"Geometry of Cydonia." could just be coincidental. He seems to find
those relationships too remarkable to be just a set of random coincidences.
Apparently, that is what his intuition about such things led him to believe.
But I have looked at all of those relationships too and my intuition told
me that they could easily be completely unintentional coincidences.
My purpose in doing the "Telephone Number Experiment" was to demonstrate
in a simple and concrete way to Hoagland and others that my intuition about
such things is quite well-founded. My choice to use eight numbers
formed from telephone numbers in the experiment was to make it obvious
that whatever relationships involving e, pi, or other mathematical
constants that exist among those numbers must be coincidental since no
one could reasonably suggest that such relationships were intentional.
I discovered many such relationships, some of which fall into rather striking
patterns. And so, if one can find an abundance of numerical
relationships from a set of numbers found in the telephone directory, why
should one put any significance on similar kinds of relationships for a
set of numbers found in some other way? I have raised this
issue a long time ago, but Richard Hoagland and his associates have failed
to offer an adequate response. Coincidences abound. One needs good,
sound arguments to believe that something is not a coincidence and so the
burden must be on Hoagland to provide such arguments, especially because
of the quite unextraordinary nature of the relationships that he presents.
It is usually impossible to prove that something is just a coincidence. For example, I cannot think of a way to definitively prove
that the architect
who designed the Great Pyramid of Giza did not really have the number pi
in mind when he chose the proportions of that structure - a popular idea
that was first suggested by John Taylor in 1859. The relationship of
pi to the Great Pyramid which was proposed by Taylor is impressively
accurate. That accuracy is the primary reason that many people have taken the
proposal seriously. But, as it turns out, that reason is rather deceptive.
See my page Pi and the
Great Pyramid for a discussion of this interesting issue.
I also cannot think of a way to definitively prove that the Torah does not intentionally include
a prediction of the assassination of the Israeli Prime Minister Rabin,
as has been suggested by Michael Drosnin . See my page The
Bible Code for a survey of this topic. Immediately after Drosnin
published his book about this, various people did "control experiments"
to show how easily one could find such "predictions" in various other texts
and documents. Such experiments are in the same spirit as my experiment
with telephone numbers.
A Martian architect could
indeed design a five-sided
pyramid incorporating the relationships proposed by Hoagland and Torun as
rough approximations. However, one of the points that I
will explain
below (in 2) is that it would not be geometrically possible to
incorporate all, or even most, of those relationships with a high degree of
accuracy. This means that if one is willing to grant the premise that an
architect who is designing a massive object such as the D&M Pyramid would strive for
a high degree of exactness in his design, then one must conclude that
many of Hoagland's mathematical relationships are in fact merely
concidental. Also, I will point out
in 5 that if one accepts a theory known as the Exploded Planet
Hypothesis, then one is led to conclude that Hoagland's very accurate
relationship relating the latitude of the D&M Pyramid to the mathematical
constant e/pi must
indeed be purely accidental - just another nice example of the ubiquity
of elegant, coincidental mathematical relationships.

2. An obvious question to ask is whether the percentage errors
occurring in the relationships given in the Torun-Hoagland Model could
turn out to be significantly better overall when (and if) more precise
measurements of the angles occurring in the D&M Pyramid are made (assuming
that it is even like a pyramid and has precise angles to be measured).
The answer is no! It would be possible for some of the percentage errors
to be better, but then others would necessarily be worse. Even if
one just considers the relationships involving the four angles A,B,C, and
D, it turns out to be impossible for all of them to be accurate.
Here are some of those relationships.

A=60oB=120oC/A=
\/2B/D=\/3A/D=e/piC/D=e/\/5C/B=\/5/pi

The source of the problem is that
the four angles A, B, C, and D are not independent. Elementary geometry
implies that there are two simple relationships between them. These
can be seen by looking at the diagram
here
where the angles are clearly indicated by colors. The two geometric
relationships are: B=2A
and B+2C+D=360o.
When
one examines the Torun-Hoagland Model with these two relationships in mind,
one finds very quickly that their relationships cannot all be accurate.
To illustrate this point, suppose for example that A=60o
and C/A=\/2
turn out to be accurate to within a percentage error of .1%. Then
B=2A would be in the range 119.88o < B < 120.12o
and D=360o-B-2C would be in the range 69.83o<
D < 70.76o . But then B/D would be a rather poor approximation
to \/3=1.73205....
The
value of B/D would be at most 1.72018 and the percentage error of the relationship
B/D=\/3
would be almost .7%.
This problem pervades the Torun-Hoagland Model. To put it simply,
the relationships that they propose in their model are contradictory.
It is impossible to design a structure for which all or even most of these
relationships are achieved to a high degree of accuracy. The
seriousness of this problem becomes more apparent if one tries to imagine
a Martian architect in the process of designing the D&M Pyramid.
Let's assume hypothetically that he wants to incorporate the two relationships
A=60o
and C/A=\/2
into his design. He would certainly realize that the angle B=2A must
be 120o and that the angle D is then completely determined and
would calculate it to be

D = 360o - B - 2C = 360o - 120o
- (2 x \/2
x 60o) = 70.29437...o

Using this value for the angle D,
it turns out that B/D=1.70710...,
which is a very poor approximation to \/3
= 1.73205.... The percentage
error is almost 1.5%. For this reason, it seems inconceivable that
the Martian architect would also have the relationship B/D=\/3
in mind. The relationship A/D=e/pi
would have a percentage error of .6%. The relationship C/D=e/\/5
would have a percentage error of more than .7%. The relationship C/B=\/5/pi
would have a percentage error of more than .6%. And so, in this hypothetical
situation, it is difficult to imagine that the Martian architect
would seriously consider any of these relationships as part of the design.
Let's consider another hypothetical possibility which turns out to be somewhat
better. The architect might want to incorporate the relationship A=60o
and B/D=\/3
into his design. Then B must again be 120o, D would
be 69.28203...o and C would be 85.35898...o.
Using this value of C, it turns out that C/A=1.42264...
and this is a poor approximation to \/2.
The percentage error is almost .6%. The relationship C/D=e/\/5
would have a percentage error of more than 1.3%. But the relationship
C/B=\/5/pi
is fairly accurate, the percentage error being just .06%. Since
A=B/2, the ratio A/D
is exactly equal to (B/D)/2
= \/3/2
and the relationship A/D=e/pi
would have a percentage error of .09%.
There are other elementary geometrical relationships between the various
angles considered by Hoagland and Torun. In addition to the
two mentioned above, one has: C+G+F=180o,
D+2H=180o, G+H=I, 2E=D.
Taking all of these geometric relationships into account, the architect
would only need to choose values for three of the above nine angles. The
remaining six angles would then be completely determined. For example,
just choosing specific values for the angles A and C would determine the
angles B, D, E and H. If the architect then made some choice of the angle
F, the angles G and I would be determined. The choice of those three angles
could conceivably be guided by a desire to incorporate a small number of
additional relationships into the design of the structure, possibly even
a few of the relationships in theTorun-Hoagland Model. Other relationships
would arise as direct consequences (such as tan(A)=\/3
if A is chosen to be 60o). And still more relationships
would arise with various degrees of accuracy as merely accidental, unintended
consequences of the choice of those three angles.
I have examined a number of hypothetical possibilities for choosing three
of the nine angles in a way which incorporates some of the relationships
in the Torun-Hoagland Model. Just as in the above illustrations,
it always turns out that many of the other relationships become quite inaccurate.
Torun and Hoagland were aware of the relationships coming from elementary
geometry (which are highlighted above in red). This is apparent from the
fact that the angles A,B,. . . ,I that they propose satisfy these relationships
exactly. But it is also obvious that Torun and Hoagland did not take
these simple geometric relationships into account in proposing the set
of relationships in their model. In my opinion, this makes their
proposed model untenable because there is no doubt that an architect sitting
at his desk to design such a massive structure would take elementary geometry
very much into account.

3.
Richard Hoagland has described the D&M Pyramid as the "Mathematical
Rosetta Stone of Cydonia." He seems to see a message encoded in all of
the mathematical constants which occur in the Torun-Hoagland Model for
that object - a message involving specifically the geometric properties
of a regular tetrahedron inscribed in a sphere. Furthermore, he claims
that this message reveals that the designers of the D&M Pyramid and
the other objects in the Cydonia region had knowledge of some kind of "hyperdimensional
physics."
I am baffled by Hoagland's interpretation of those mathematical constants.
Even if it were somehow known that the D&M Pyramid had really been
designed by some intelligent beings and that all of the relationships that
he and Torun have found were really intentionally incorporated into its
design (overlooking the discussion in 2), his interpretation would
make no sense to me. The problem is that the various mathematical constants
that occur in those relationships don't support the interpretation that
Hoagland is imposing on them.
I don't see any connection between the geometry of a tetrahedron and constants
such as e/pi, e/\/5,
and pi/\/5
(or their reciprocals). There are many constants which play
a very important role in mathematics. Certainly, the numbers e , pi,
and square roots such as \/5
are of critical importance throughout mathematics. But ratios of
such numbers are not necessarily important. The ratio e/pi,
which is a number that Hoagland especially singles out, seems somewhat
boring to me. I cannot think of any formula or theory where that
specific number occurs naturally. Of course, some ratios are indeed
important numbers, e.g. the ratio \/3/2
occurs often in geometry, as do both \/3
and 2. I have no doubt that I could easily find some nice geometric
interpretation of the ratio pi/\/5
involving the geometry of Platonic solids and spheres. I could probably
find something interesting to say about the ratios e/\/5
and e/pi , but nothing
which would make those numbers worthy of being incorporated in a monument.
Some of the other constants which occur in the Torun-Hoagland Model (e.g.
pi/3, 2pi/3,
and \/3
) are connected with the geometry of an equilateral triangle. Since
the four faces of a regular tetrahedron are equilateral triangles, this
would perhaps be an indirect connection with the geometry of a regular
tetrahedron. But equilateral triangles occur in a countless number of ways
in various designs and geometric objects, of which the regular tetrahedron
is just one example.
Hoagland and Torun wrote an article in 1989 (the "Message
of Cydonia") where they argue that the constant e/pi
is the key and that if one replaces e=2.7182818... by the constant
e'=(\/3/2)pi
= 2.7206990... in those relationships (so that e/pi
is replaced by e'/pi),
then one has a compelling connection with the geometry of a tetrahedron
inscribed in a sphere. It is certainly true that this constant
e' has very interesting connections with that geometry. Hoagland
and Torun point out the following connection. If a regular tetrahedron
T
is inscribed in a sphere S, then it turns out that the ratio of
the surface areas of the sphere and the tetrahedron is exactly equal to
e':

e' = (Surface Area of S)/(Surface
Area of T) ..

Here is another connection. If two regular tetrahedra
of the same size are inscribed in a sphere so that each vertex of one is
diametrically opposite to a vertex of the other, then the eight vertices
(four vertices from each tetrahedron) form the vertices of a cube C
inscribed in the sphere. Then the number e' turns out to be exactly equal
to the ratio of the volumes of the sphere and the cube:

e' = (Volume of S)/(Volume
of C).

But this interesting ("tetrahedral") constant e'
does not actually occur in any significant way in the numerical relationships
found by Hoagland and Torun. It is not the number that Hoagland and Torun
single out as the key. The number that they single out is \/3/2.
It is this number that occurs a few times in the Torun-Hoagland Model ("redundantly",
as Hoagland likes to say). The link between these two numbers is
that they are related by a factor of pi.:..e'.=.(\/3/2)pi.
Now the number \/3/2
occurs in quite a few contexts and would itself bring to mind many associations
to anyone knowledgeable about geometry (without any need to multiply it
by the number pi). Geometry related to a tetrahedron inscribed
in a sphere would be somewhere down the list. Here are some
of the associations for \/3/2
that come to my mind: (1) It's equal to sin(60o) and cos(30o).
(2) It's the length of an altitude of an equilateral triangle whose side
has length 1. (3) It's the ratio of the radius of the circle
inscribed in a given regular hexagon to the radius of the circle circumscribing
the regular hexagon. (4) It's the radius of the circle circumscribing
a rectangle whose sides are of length 1 and \/2.
(This rectangle will be mentioned later in connection with the "Mounds"
of Cydonia.) (5) It's the radius of the sphere which circumscribes
a cube with side of length 1. Only the last one in this list has
a connection with the geometry of a tetrahedron inscribed in a sphere,
a somewhat indirect connection resulting from the fact that the eight vertices
of a cube turn out to be the vertices of two regular tetrahedra (each of
which has four vertices) and the sphere which circumscribes a cube also
circumscribes those two regular tetrahedra.
Richard Hoagland has lost me in the giant leap that he makes from the constants
occurring in the Torun-Hoagland Model to the idea that, sometime in the
past, a Martian architect decided to encode a subtle message about tetrahedral-spherical
geometry in the design of a five-sides pyramid on his planet. It
is an interesting story, but doesn't correspond very well to those constants.
I can see only an extremely flimsy connection. But then he
loses me completely by making another, even larger, giant leap, suggesting
that this encoded tetrahedral-spherical geometry (which, for me, is just
part of old-fashioned Euclidean three-dimensional geometry) is really a
message encoded by that remarkable architect about some kind of Physics
involving more than three dimensions and that this Physics furthermore
can somehow explain the locations of volcanos on Earth and other planets
as well as the Dark Spots on Jupiter and Neptune.

4.
Hoagland's elaborate interpretation seems absurd for other reasons too.
With the generous level of accuracy that Hoagland and Torun give themselves,
there are just too many numerical relationships. The very number
and variety of such numerical relationships would make it impossible to
find any clear message. It would be almost like a mathematical Rorschach
test. Any "messages" found in the numbers would be a reflection
of the mindset of the individual taking the test. Just as a simple
illustration, consider the rather striking set of relationships involving
the numbers \/2,..\/3,..\/4,..\/5,..\/6,..\/7
which I presented above. Those numbers might suggest the curve y=\/x.
This curve is part of a parabola and that might bring to mind many associations.
Instead of finding a message involving "hyperdimensional physics," as Hoagland
does, perhaps some other individual might regard these relationships as
the basis of a message involving classical Newtonian physics, where parabolas
occur in a variety of ways. That interpretation is just one of many
that could be imposed on those numbers. It is certainly as tenable as Hoagland's
interpretation, perhaps even more so in light of the discussion in
3.
As an illustration of how easy it is to impose an interpretation on a bunch
of numbers, the reader should visit Tetrahedral Geometry
in the Seattle Telephone Directory. As I was writing the page
describing my Telephone
Number Experiment, I decided to look for connections with the geometry
of a regular tetrahedron inscribed in a sphere (motivated by Hoagland's
apparent fascination with that geometry). It was rather amusing to
find that the numbers that come up in that experiment (as ratios, etc.)
seem to have an even more compelling connection with that geometry than
the numbers occurring in the Torun-Hoagland Model for the D&M Pyramid.
Even the numbers themselves that occur in the Torun-Hoagland Model are
ambiguous. For example, as Hoagland and Torun are aware, the
numbers e/pi
and \/3/2
(which is e'/pi) are quite
close to each other. As another example,
\/3 and 1/gamma are also
quite close to each other. Thus, in any
numerical relationship involving one of these numbers, one could also give
another numerical relationship involving the other number, with only a
slightly different percentage error. If these relationships are intentional,
which of the numbers is what the architect had in mind?
Any artificial structure will exhibit a large number of unintended relationships
in addition to whatever intended relationships might have been chosen by
the builders or architects. The Great Pyramid of Giza is an example. One
can find quite a number of fairly accurate numerical relationships exhibited
by the Great Pyramid. (See my page Some
Elegant Numerical Relationships.) It seems likely that only one (or
at most one) of those relationships was actually intentional. With an object
like the D&M Pyramid, even if one assumes that it is artificial, how
can one realistically guess the intentions of the architect. Those
intentions might have absolutely nothing to do with ratios of angles.
Perhaps the architect was more concerned with relationships involving lengths
or areas or volumes associated with the object. Such relationships are
also bound to occur. (See my page Secrets
of a Certain Pentagon for a description of another numerical experiment
illustrating this point.) Even if the architect's intentions
did involve ratios of angles, only a subset of all of those relationships
pointed out by Hoagland and Torun, or the additional ones pointed out by
me, or others that no one has yet noticed could be intentional (as I have
argued in 2 above). But which subset would that be from among
the incredibly large number of hypothetical possibilities?
I am really making two objections here. First of all, among the plethora
of relationships that one could find in any artificial object (even with
a fairly high degree of accuracy), it is virtually impossible to
decide which ones are intentional (unless there is solid documentation
available concerning the design). Second of all, even if it
were certain that specific mathematical constants were intentionally incorporated
in the design, how can one possibly guess what those numbers meant to the
architect without knowing much more about the ambient culture and its traditions.

5. The
above comments have concentrated on the Torun-Hoagland Model for the D&M
Pyramid, which is just part of what Richard Hoagland calls the "Geometry
of Cydonia." In The Monuments of Mars , Hoagland
discusses various alignments of objects such as the D&M Pyramid, the
Face, etc. He discusses at length the probabilities of those specific
alignments occurring just at random. He points out various occurrences
of angles such as 19.5o (which is Hoagland's favorite angle
and represents arcsin(1/3)=19.4712206...o). He points
out that the latitude of the D&M Pyramid is related to e/pi.
He points out various triangles which are formed by taking certain formations
(the so-called "mounds" ) as vertices.
The kinds of probability calculations that Hoagland presents are meaningless.
He and his associates and followers seem to make this mistake over and
over again. Tom Van Flandern explains the issue very well in an essay that
I will briefly discuss later.

"...... In any truly random data set, many regular
patterns can always be found. For example, if we have a star chart with
a million stars, we might find an unusual shape formed by stars that has
less than one chance in a billion of happening by chance. So are some mysterious
super-beings moving stars around? This is not as likely as the simpler
explanation: In every random data set capable of forming billions of random
patterns, it is virtually certain that some 1-in-1 billion pattern will
be found formed by chance.

In
general, we tend to be deceived because our minds often do not recognize
how truly vast is the number of possible coincidences that can occur. So
when a few of them do occur, as they must if the odds are right, we tend
to be amazed simply because the odds against that particular coincidence
were very great. The odds against a flipped coin coming up tails ten straight
times are 1024-to-1 against. But if we make several thousand attempts,
the odds become pretty good that it will happen one or more times.

In science, an improbable event that has already happened is called
"a posteriori" (after the fact), and generally is taken to have no significance
no matter how unlikely it might appear. By contrast, if we specified a
certain specific highly improbable event in all its detail "a priori" (before
the fact), and it happened anyway, that would be significant, and we would
be obliged to pay attention."

The last paragraph, in particular, is quite relevant.
All of the probabilities that Hoagland calculates are "a posteriori," and
so have no real significance. Improbable events are probable! That seemingly
paradoxical statement is illustrated fairly well in some of the experiments
that I have described elsewhere on this site and referred to above.
Just to choose one of many examples, consider the two relationships
mentioned on my page Secrets
of a Certain Pentagon involving the Golden Section phi. One
of those relationships is completely intentional, just part of the way
the pentagon considered on that page is constructed. But the other relationship
is unintentional - an accidental consequence of Horace Crater's random
choice of the angle D. The accuracy of that relationship is impressive
and the probability for that specific relationship occurring so accurately
is extremely small. But since I did not make an a priori choice
to look for that specific relationship, or even the Golden Section as a
specific number, there is nothing miraculous about finding such an extremely
improbable relationship.As far as
the occurrence of the angle 19.5o here and there throughout
the Cydonia region, this is fully to be expected because of the incredibly
large number of angles that are formed by various objects that Hoagland
and others have singled out for attention. To give the reader some
intuition about this, suppose that one considers just 12 points. If you
draw all possible line segments between such points, you will have 66 such
line segments. The number of triangles that you can form using these line
segments as sides (i.e. using any three of the 12 points as vertices)
turns out to be 220. Each such triangle has three angles and so one
gets a total of 660 angles. (If it happens that three points are on a single
line, then some of the angles may coincide.) If instead one starts
from 25 points, then it turns out that one can form 2300 triangles and
therefore one gets a total of 6900 angles, all between 0o and
180o. In fact, Hoagland has more than 25 points that he
could potentially have used as vertices to form angles. With so many
possibilities, it is almost inevitable that quite a number of these angles
will be very close to 19.5o, 45o, 60o
or any other angle that one might be interested in looking for. I
cannot see any significance in the fact that Hoagland has found several
19.5o angles, especially because they just occur scattered about
and form no systematic or symmetrical pattern. Hoagland finds
significance in the fact that the (hypothetical) apex of the D&M Pyramid
is located at a latitude of approximately 40.87o N.
As he points out, tan(40.87o) is approximately e/pi. The
level of accuracy is quite high. Undoubtedly, Hoagland carefully
looked at the latitudes of the other objects in the Cydonia region.
Since he mentions nothing about the latitude of the Face itself, it would
seem that he found nothing that he considered significant about it. The
latitude of the center of the Face, which I will denote by the letter L,
is between 41.18o N and 41.19o
N
(based
on a map in Hoagland's book). Actually, there is a very nice
relationship for that latitude too: tan(L)=7/8. The level of accuracy is
again quite high. If the latitude of the D&M Pyramid is significant,
why shouldn't the latitude of the Face also be significant? Just
as with the angle-relationships for the D&M Pyramid, Hoagland seems
to be rather selective - emphasizing some relationships, and ignoring or
not noticing others. There
is a theory that the axis of rotation of Mars has dramatically shifted
over time and that some shift has even taken place in the relatively recent
past. One can find some papers in well-known scientific journals
on this topic. Such a shift would change the latitude of almost everything
on the surface of Mars. Tom Van Flandern has a theory that Mars was once
a moon of a planet that was destroyed by an explosion. A dramatic
shift in the axis of rotation would be an almost inevitable consequence
of such a catastrophic event. Van Flandern suggests that the Face
may have once been in the vicinity of the Martian equator before such an
explosion. If Van Flandern is right (and Hoagland seems to be quite
supportive of Van Flandern's theory), then it would seem that the
present location of the D&M Pyramid at a latitude of 40.87o
N is merely an accidental consequence of natural events. If
so, then both of the above very accurate relationships involving that latitude
as well as the latitude of the Face would be nothing more than coincidences-
more illustrations of the "Power of Randomness" (as are all of the other
relationships pointed out by Hoagland in my opinion).

When I challenged Richard Hoagland to a debate back in May, 1998, I stated
my position as follows: "You have often stated that the
numerical relationships exhibited by various objects on Cydonia provide
strong evidence that these objects are artificial. My contention is that
virtually all of this evidence is fallacious and should be discarded."
The reason that I used the word "virtually"in my statement is because of
a paper by Horace Crater and Stanley McDaniel entitled Mound Configurations
on the Martian Cydonia Plain. At that time, I had looked at their
paper, but had not thoroughly studied it. My initial reaction was a combination
of skepticism about their approach and curiosity about how strong their
evidence really was. On the surface, it seemed that their evidence might
possibly be quite interesting. During June, 1998, I carefully studied
their paper. It might be worthwhile for me to discuss my opinion
of their work which, as the reader will see, turns out to be somewhat inconclusive.
For that, turn to another page:
Cydonia Mound Geometry